Mic.2
Consider a system of N independent harmonic oscillators. The energy of each
oscillator is quantised and is given, for the $i^{th}$ oscillator, by $E_i = \epsilon(n_i + \frac{1}{2})$,
with $n_i = 0, 1, 2, ...$
(a) Find the number of accessible states $\Gamma(n, N)$, where the energy is fixed,
$E = \epsilon(n + N/2)$, and $n = n_1 + n_2 + ... + n_N$.
(b) Find the entropy of the system, $S(n, N)$, in an approximation where
$N, n \gg 1$.
(c) Compute the temperature T of the system at equilibrium. Deduce an
expression for $S(T, N)$.
(d) Find the free energy of the system, $F(T, N)$.
N.B. It is known from thermodynamics that $F = E - TS$.